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We consider a non-resonant system of finitely many bilinear Schroedingerequations with discrete spectrum driven by the same scalar control. We provethat this system can approximately track any given system of trajectories ofdensity matrices, up to the phase of the coordinates. The result is valid bothfor bounded and unbounded Schroedinger operators. The method used relies onfinite-dimensional control techniques applied to Lie groups. We provide also anexample showing that no approximate tracking of both modulus and phase ispossible

Simultaneous approximate tracking of density matrices for a system of Schroedinger equations